CAREER: Nonlinear PDE Models in Mathematical Physics and Experiment

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Abstract Understanding interesting physical phenomena often requires classifying potentially physical observable solutions as attractive critical points (or semi-stable long-lived orbits) of infinite dimensional dynamical systems through partial differential equation theory and numerical experiments, which provide a rich set of problems that can be accessible at all levels of research and training. Such solutions and their stability can be studied on long time scales in relation to small scale models for light in optical media in nonlinear Schrodinger and Dirac models, surface waves on a surface tension scale in the gravity-capillary equations, or molecular dynamics in terms of both large scale nonlinear diffusions of crystals through thermodynamic fluctuations as well as Lagrangian mechanics for smaller systems of electrons trapped in various nuclear potentials. They can also be studied in macroscopic systems such as interaction of surface waves and internal waves in coupled fluids models, vortex formation in fluid flow around biological objects in the Navier-Stokes equations with boundary, and dark matter formation in models from general relativity using Einstein-Scalar Field equations and their reductions as Schrödinger-Poisson models. In applications that include large systems, complicated nonlinearities, and/or interesting geometric settings, the analysis and numerics can become increasingly difficult. The quest to understand complexity in partial differential equation models has led to the development of dramatically new analytic and computational techniques to explore questions of symmetry, phase transitions, uniqueness for the attractive states, as well as generalizations of Fourier transform methods using scattering theory, spectral theory and microlocal analysis to understand their stability. These techniques can be applied for instance on spaces with curvature, boundary, metric singularities or other difficult features such as noise in a sample or trapping due to external potential wells. This proposal will involve research in partial differential equations directly related to optics and electronic structure with relevant boundary conditions and potentials, as well as other equations from fluid dynamics, general relativity and thermodynamic fluctuations on crystal surfaces. The PI will also work with postdoctoral fellows, graduate students and undergraduates on integrated research into models, computation and experiment, especially though collaboration with members of the UNC Fluids Lab and International Mathematics Climate Network. An important aspect of that training will be to develop graduate courses and undergraduate courses in dynamics and computation to prepare trainees for a variety of careers in science. From a human resources standpoint, models like those in this proposal provide a large pool of problems that give a strong background in computation and geometry, as well as some applied statistics, which can be used for training purposes in work with researchers of all levels, from undergraduate to postdoctoral, then applied in many scientific fields. In addition, the PI will continue to support the mathematics department role in the University of North Carolina Science Expo to work towards broader outreach goals of making mathematical sciences more accessible to the public.